A simple reformulation of Riemann Zeta function

0.07 MB·English

by  Riccardo Poli

A simple reformulation of Riemann’s Zeta function
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Riccardo Poli and W. B. Langdon
0
0 Department of Computer Science
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University of Essex,

n
Wivenhoe Park, Colchester, CO4 7SQ, UK
a
J {rpoli,wlangdon}@essex.ac.uk

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] Abstract
O
H Werewrite Riemann’s Zeta function as
h. ζ(s)=1+Xp−sY 1−1q−s
t p q≥p
a
m where p and q are primes.
[

1 1 Introduction
v
0 ∞
6 TheconnectionbetweenRiemann’szetafunctionζ(s)= n−s[2]andEuler’sproductformula[1]
1 nX=1
1 haslongbeenknown. Inthis paper weintroduceanintermediaterepresentationforζ(s) asasum
0 of products over the primes.
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0
/ 2 Results
h
t
a
m Definitions. Let {pk} be the sequence of the primes,
: i
v
Z (s)= (1−p−s)−1,
i i k
X kY=1
r
a
i i
Si(s)= p−ks (1−p−js)−1
Xk=1 jY=k
 
and
I ={s∈C|p−s =1 for some prime p≤p }.
i i
Explicitly
(1+2k)π
I = s∈C ℜ(s)=0∧ ∃k ∈Z,j ∈{1,··· ,i} : ℑ(s)= .
i
(cid:26) (cid:12) (cid:18) lnpj (cid:19)(cid:27)
(cid:12)
(cid:12)
Theorem. For any i∈N and s∈C\I , we have Z (s)=1+S (s).
i i i
Proof: We proceed by induction.
Base case: For i=1 we have
S1(s)=p−1s(1−p−1s)−1 ⇒
1−p−s+p−s 1
1+S1(s)=1+p−1s(1−p−1s)−1 = 1−1 p−s 1 = 1−p−s =Z1(s).
1 1
1
Induction hypothesis: Assume
Z (s)=1+S (s).
i i
By adding and subtracting p−s to the r.h.s. we obtain
i+1
Z (s) = p−s +S (s) + 1−p−s ⇒
i i+1 i i+1
(cid:0) (cid:1) (cid:0) (cid:1)
(1−p−s )−1Z (s) = (1−p−s )−1 p−s +S (s) +(1−p−s )−1(1−p−s )⇒
i+1 i i+1 i+1 i i+1 i+1
(cid:0) (cid:1)
i i
Zi+1(s) = (1−p−i+s1)−1p−i+s1+ p−ks (1−p−js)−1+1
Xk=1 jY=k
  
i i+1
= (1−p−s )−1p−s + p−s (1−p−s)−1+1
i+1 i+1 k j
kX=1 jY=k
 
i+1 i+1
= p−s (1−p−s)−1+1⇒
k j
kX=1 jY=k
 
Zi+1(s)=1+Si+1(s).
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Corollary. For ℜ(s)>1, Riemann’s Zeta function can be written as
∞
ζ(s)=1+ a (s)p−s
k k
kX=1
∞
where a (s)= (1−p−s)−1.
k j
jY=k
Proof: The result follows directly from noting that Euler’s product formula can be expressed as
ζ(s)= lim Z
i
i→∞
and from the previous theorem. 2
3 Conclusions
In our reformulationfor the Riemann’s Zeta function, ζ(s) is expressedas a sum over the primes,
where eachterm is a product with factors depending only on the summation index (a prime) and
the primes following it. The formulation is interesting because of its similarity with the original
∞
definition of the Zeta function, ζ(s)= n−s, the factors a (s) effectively being correctionsthat
k
nX=1
account for all the (non-prime) naturals whose prime factor decomposition starts with p .
k
Acknowledgements
The authors would like to thank John Ford and Ray Turner for helpful discussions.
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References
[1] Leonard Euler. Introductio in Analysin Infinitorum. Bosquet, Lucerne, 1748.
[2] BernhardRiemann. U¨berdieAnzahlderPrimzahlenuntereinergegebenenGr¨osse.Monatsber.
Ko¨nigl. Preuss. Akad. Wiss, pages 671–680,November 1859.
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